On Edge-Disjoint Empty Triangles of Point Sets

Thirty Essays on Geometric Graph Theory ◽

10.1007/978-1-4614-0110-0_7 ◽

2012 ◽

pp. 83-100 ◽

Cited By ~ 1

Author(s):

Javier Cano ◽

Luis F. Barba ◽

Toshinori Sakai ◽

Jorge Urrutia

Keyword(s):

Point Sets ◽

Edge Disjoint ◽

Empty Triangles

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Planar point sets with a small number of empty convex polygons

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.41.2004.2.4 ◽

2004 ◽

Vol 41 (2) ◽

pp. 243-269 ◽

Cited By ~ 5

Author(s):

Imre Bárány ◽

Pável Valtr

Keyword(s):

Convex Hull ◽

General Position ◽

Point Sets ◽

Convex Polygons ◽

Planar Point ◽

Empty Convex ◽

Finite Set ◽

Empty Triangles

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ˜1.62n2 empty triangles, ˜1.94n2 empty quadrilaterals, ˜1.02n2 empty pentagons, and ˜0.2n2 empty hexagons.

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Monochromatic empty triangles in two-colored point sets

Discrete Applied Mathematics ◽

10.1016/j.dam.2011.08.026 ◽

2013 ◽

Vol 161 (9) ◽

pp. 1259-1261 ◽

Cited By ~ 6

Author(s):

János Pach ◽

Géza Tóth

Keyword(s):

Point Sets ◽

Empty Triangles ◽

Colored Point

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Representation of certain self-similar quasiperiodic tilings with perfect matching rules by discrete point sets

Journal de Physique I ◽

10.1051/jp1:1994234 ◽

1994 ◽

Vol 4 (6) ◽

pp. 893-904 ◽

Cited By ~ 4

Author(s):

Richard Klitzing ◽

Michael Baake

Keyword(s):

Perfect Matching ◽

Discrete Point ◽

Point Sets ◽

Self Similar ◽

Matching Rules ◽

Quasiperiodic Tilings

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Almost empty polygons

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.40.2003.3.1 ◽

2003 ◽

Vol 40 (3) ◽

pp. 269-286 ◽

Cited By ~ 4

Author(s):

H. Nyklová

Keyword(s):

Exact Results ◽

Point Sets ◽

Planar Point ◽

Convex Position ◽

Vertex Set ◽

Point Set ◽

Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

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